Euler–Riemann Zeta Function

A Euler–Riemann Zeta Function is a Zeta function of a complex variable.

• Context:
  • It can (often) be defined as the analytic continuation of the sum of the infinite series \(\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\) for complex numbers \(s\) with real part greater than 1.
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  • It can range from simple forms such as [math]\displaystyle{ \(\zeta(2)\) which equals \(\frac{\pi^2}{6}\) }[/math] to complex cases involving non-trivial zeros on the critical line.
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  • It can be extended to the entire complex plane, except for a simple pole at \(s=1\).
  • It can be connected to the distribution of prime numbers through the Riemann Hypothesis, a conjecture that posits all non-trivial zeros of the zeta function have a real part equal to \(\frac{1}{2}\).
  • It can exhibit functional symmetry through the functional equation, relating \(\zeta(s)\) to \(\zeta(1-s)\).
  • It can also be expressed using integral representations, particularly through the Mellin transform.
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• Example(s):
  • An example of its role in prime number theory is the Prime Number Theorem, which approximates the distribution of primes by using the non-trivial zeros of the zeta function.
  • An example of its functional equation [math]\displaystyle{ \(\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)\) }[/math].
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• Counter-Example(s):
  • Dirichlet Eta Function, ... of the Euler–Riemann Zeta Function.
  • Hurwitz Zeta Function, ... of the Euler–Riemann Zeta Function.
• See: L-Function, Riemann Hypothesis, Complex Variable, Analytic Continuation, Analytic Number Theory, Physics, Probability Theory, Statistics, Leonhard Euler, Real Numbers.

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Riemann_zeta_function Retrieved:2024-8-26.
  • The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as [math]\displaystyle{ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots }[/math] for and its analytic continuation elsewhere.

    The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics.

    Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics.

    The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet -functions and -functions, are known.

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